A light rope is wound around a hollow cylinder of mass \(5\,kg\) and radius \(70\,cm\). The rope is pulled with a force of \(52.5\,N\). The angular acceleration of the cylinder will be.....\( {rad \,s}{ }^{-2}\).

In the experimental setup of meter bridge shown in the figure, the null point is obtained at a distance of \(40\,cm\) from \(A\). If a \(10\,\Omega \) resistor is connected in series with \(R_1\), the null point shifts by \(10\,cm\). The resistance that should be connected in parallel with \(\left( {{R_1} + 10} \right)\,\Omega \) such that the null point shifts back to its initial position is .............. \(\Omega\)

In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

An massless equilateral triangle \(EFG\) of side \('a'\) (As shown in figure) has three particles of mass \(m\) situated at its vertices. The moment of intertia of the system about the line \(EX\) perpendicular to \(EG\) in the plane of \(EFG\) is \(\frac{ N }{20}\, ma ^{2}\) where \(N\) is an integer. The value of \(N\) is

A simple harmonic oscillator of angular frequency \(2\,rad\,s^{-1}\) is acted upon by an external force \(F = sin\,t\,N .\) If the oscillator is at rest in its equilibrium position at \(t = 0,\) its position at later times is proportional to

Eight mercury drops, each of radius \(r\), coalesce to form a bigger drop. The surface energy released in this process is _______. (\(S\) is the surface tension of mercury).

When a coil is placed in a time dependent magnetic field the power dissipated in it is \(P\). The number of turns, area of the coil and radius of the coil wire are \(N\), \(A\) and \(r\) respectively. For a second coils number of turns, area of the coil and radius of the coil wire are \(2N\), \(2A\) and \(3r\) respectively. When the first coil is replaced with second coil the power dissipated in it is \(\sqrt{2}\,\alpha P\). The value of \(\alpha\) is _______.

\(\mathrm{N}\) moles of a polyatomic gas \((f=6)\) must be mixed with two moles of a monoatomic gas so that the mixture behaves as a diatomic gas. The value of \(\mathrm{N}\) is _______.

A uniform solid cylindrical roller of mass \('m'\) is being pulled on a horizontal surface with force \(F\) parallel to the surface and applied at its centre. If the acceleration of the cylinder is \('a'\) and it is rolling without slipping then the value of \('F'\) is 

An asteroid is moving directly towards the centre of the earth. When at a distance of \(10 \mathrm{R}\) (\(R\) is the radius of the earth) from the earths centre, it has a speed of \(12 \;\mathrm{km} / \mathrm{s}\). Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is \(11.2 \;\mathrm{km} / \mathrm{s}\) ) \(?\) Give your answer to the nearest integer in \(km/s\)